QJEEM22Mae8cf: JEE Main 2022 Physics - Class 11 JEEM22Mae8cf Solution: Let a vector has magnitude 9. Let a vector be such that for…

$\left( {x\overrightarrow a + y\overrightarrow b } \right).\left( {6y\overrightarrow a - 18x\overrightarrow b } \right) = 0$

$\Rightarrow \left( {6xy|\overrightarrow a {|^2} - 18xy|\overrightarrow b {|^2}} \right) + \left( {6{y^2} - 18{x^2}} \right)\overrightarrow a .\overrightarrow b = 0$

As given equation is identity

Coefficient of ${x^2} =$ coefficient of ${y^2} =$ coefficient of $xy = 0$

$\Rightarrow |\overrightarrow a {|^2} = 3|\overrightarrow b {|^2} \Rightarrow |\overrightarrow b | = 3\sqrt 3$

and $\overrightarrow a .\overrightarrow b = 0$

$|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a ||\overrightarrow b |\sin \theta$

$= 9.\,3\sqrt 3 .1 = 27\sqrt 3$